**ABSTRACT**
### Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless
Systems

#### David Love, Robert Heath, and Thomas Strohmer

Transmit and receive beamforming is an
attractive, low-complexity technique for exploiting the
significant diversity that is available in multiple-input and
multiple-output (MIMO) wireless systems. Unfortunately, optimal
performance requires either complete channel knowledge or
knowledge of the optimal beamforming vector which is difficult to
realize in practice. In this paper, we propose a quantized maximum
signal-to-noise ratio beamforming technique where the receiver
only sends the label of the best weight vector in a predetermined
codebook to the transmitter. We develop optimal codebooks for
i.i.d. Rayleigh fading matrix channels. We derive the distribution
of the optimal transmit weight vector and use this result to bound
the SNR degradation as a function of the quantization. A codebook
design criterion is proposed by exploiting the connections between
the quantization problem and Grassmannian line packing. The design
criterion is flexible
enough to allow for side constraints on the codebook vectors. Bounds on
the maximum distortion with the resulting Grassmannian codebooks follow
naturally
from the nature of the code. A proof is given that any system using an
overcomplete
codebook for transmit diversity and maximum ratio combining obtains a
diversity on the
order of the product of the number of transmit and receive antennas.
Bounds on the loss
in capacity due to quantization are derived. Monte Carlo simulations are
presented that compare
the symbol error probability for different quantization strategies.

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